Last edited by Shaktilrajas
Sunday, August 2, 2020 | History

6 edition of Compact Riemann Surfaces (Lectures in Mathematics. ETH Zürich) found in the catalog.

# Compact Riemann Surfaces (Lectures in Mathematics. ETH Zürich)

## by R. Narasimhan

Written in English

Subjects:
• Calculus & mathematical analysis,
• Differential & Riemannian geometry,
• Riemannian Manifolds,
• Mathematics,
• Science/Mathematics,
• Calculus,
• Mathematics / Calculus,
• Mathematics / General,
• General,
• Riemann surfaces

• The Physical Object
FormatPaperback
Number of Pages132
ID Numbers
Open LibraryOL9090057M
ISBN 103764327421
ISBN 109783764327422

Instead we will concentrate on those topics which lead to a good understanding of compact Riemann surfaces. Sections of the book where we concentrate our attention: 1, 2, 4, 6, , , , 29, with 25 and 26 only touched upon lightly and 17 presented from quite a different point of view. This volume is an introduction to the theory of Compact Riemann Surfaces and algebraic curves. It gives a concise account of the elementary aspects of different viewpoints in curve theory. Foundational results on divisors and compact Riemann surfaces are also stated and proved.

For this and most other things about Riemann surfaces, I recommend Donaldson's Notes on Riemann surfaces, which are based on a graduate course I was once lucky enough to see, and which may eventually make it into book format.. In his account, the "main theorem for compact Riemann surfaces" says that one can solve $\Delta f = \rho$ for any 2-form $\rho$ with integral zero. Only a number theorist (like me) would cast Peter Buser’s Geometry and Spectra of Compact Riemann Surfaces in a supporting role vis à vis number theoretic themes. In point of fact the book is, by the author’s own description, devoted to “two subjects [:] the geometric theory of compact Riemann surfaces of genus greater than one [i.e.

one-forms on the compact Riemann surface. Eventually, it will turn out that the genus is a topological invariant. Much of this chapter is concerned with the existence of meromorphic functions on compact Riemann surfaces with prescribed principal parts or divisors. Non-compact Riemann surfaces are at the focus of chapter 6. The function the-.   This is an introduction to the geometry of compact Riemann surfaces, largely following the books Farkas-Kra, Fay, Mumford Tata lectures. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations. 2) Space of meromorphic functions and forms, we classify them with the Newton polygon. 3) Abel map, the Jacobian and Theta functions. 4) The Riemann .

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### Compact Riemann Surfaces (Lectures in Mathematics. ETH Zürich) by R. Narasimhan Download PDF EPUB FB2

Compact Riemann Surfaces has been added to your Cart Add to Cart. Buy Now More Buying Choices 9 New from $10 Used from$ 19 used & new from $See All Buying Options Available at a lower price from other sellers that may not offer free Prime by: Although Riemann surfaces are a time-honoured field, this book is novel in its broad perspective that systematically explores the connection with other fields of : Springer-Verlag Berlin Heidelberg. The book is a very useful reference for researches and also for graduate students interested in the geometry of compact Riemann surfaces of constant curvature -- 1 and their length and eigenvalue spectra.” (Liliana Răileanu, Zentralblatt MATH, Vol. ) “Geometry and Spectra of Compact Riemann Surfaces is a pleasure to by: Further Properties of Compact Riemann Surfaces. *immediately available upon purchase as print book shipments may be delayed due to the COVID crisis. ebook access is temporary and does not include ownership of the ebook. Only valid for books with an ebook version. Springer Reference Works are not : Birkhäuser Basel. Another excellent analytic monograph from this point of view is the Princeton lecture notes on Riemann surfaces by Robert Gunning, which is also a good place to learn sheaf theory. His main result is that all compact complex one manifolds occur as the Riemann surface of an algebraic curve. Introduction Although Riemann surfaces are a time-honoured field, this book is novel in its broad perspective that systematically explores the connection with other fields of mathematics.$\begingroup$I know Forster's book quite well, having taught out of a good portion of it a few times. It is extremely well-written, but definitely more analytic in flavor. In particular, it includes pretty much all the analysis to prove finite-dimensionality of sheaf cohomology on a compact Riemann surface. The complex plane C is the most basic Riemann surface. The map f(z) = z (the identity map) defines a chart for C, and {f} is an atlas for map g(z) = z * (the conjugate map) also defines a chart on C and {g} is an atlas for charts f and g are not compatible, so this endows C with two distinct Riemann surface structures. In fact, given a Riemann surface X and its atlas A, the. Search within book. Front Matter. Pages i-v. PDF. Algebraic functions. Raghavan Narasimhan. Pages Riemann Surfaces. Raghavan Narasimhan. Pages Further Properties of Compact Riemann Surfaces. Raghavan Narasimhan. Pages Hyperelliptic Curves and the Canonical Map. Raghavan Narasimhan. Pages Although Riemann surfaces are a time-honoured field, this book is novel in its broad perspective that systematically explores the connection with other fields of mathematics. a compact Riemann surface are determined almost completely when the order of the function is given at each point of the surface; and that is done most conveniently in terms of divisors. In general a divisor on a Riemann surface M. An initially more general looking deﬂnition of hyperelliptic Riemann surfaces is the class of compact Riemann surfaces M that have a holomorphic map ’: M. S2 that is a two-to-one branched cover. Actually, this is the same class as described above. To see this, suppose the branch points in M are fpj: 1 • j • V g, branching over fej: 1. J.-B. Bost, Introduction to compact Riemann surfaces, Jacobians, and abelian varieties, in From number theory to physics (Les Houches, ), Springer, Berlin,pp. It is clearly written, contains historical comments and a lot of mathematical gems. This classic monograph is a self-contained introduction to the geometry of Riemann surfaces of constant curvature -1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces. It is particularly pleasing that the subject of Riemann surfaces has attracted the attention of a new generation of mathematicians from (newly) adjacent fields (for example, those interested in hyperbolic manifolds and iterations of rational maps) and young physicists who have been convinced (certainly not by mathematicians) that compact 5/5(1). In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann particular it implies that every Riemann surface admits a Riemannian metric of constant compact Riemann surfaces, those with universal cover the unit disk are. Jürgen Jost, the author of Compact Riemann Surfaces: An Introduction to Contemporary Mathematics, characterizes his book as, among other things, “an introduction to non-linear analysis in geometry.”This is a particularly tantalizing phrase in the wake of all the excitement about Perelman’s proof of the Poincaré conjecture, accompanied as it was by everything from high drama to low comedy. 58 2. COMPACT RIEMANN SURFACES In this chapter we study compact Riemann surfaces in some detail and obtain several important results. We start (§) with the deﬁnition of divisor: loosely speaking a divisor is a ﬁnite set of points of a surface to which certain integral weights (numbers) are assigned. Lectures notes on compact Riemann surfaces by Bertrand Eynard. Publisher: Number of pages: Description: This is an introduction to the geometry of compact Riemann surfaces. Free shipping on orders of$35+ from Target. Read reviews and buy Riemann Surfaces - (Graduate Texts in Mathematics) 2nd Edition by Hershel M Farkas & Irwin Kra (Hardcover) at Target.

Get it today with Same Day Delivery, Order Pickup or Drive Up. This book establishes the basic function theory and complex geometry of Riemann surfaces, both open and compact. Many of the methods used in the book are adaptations and simplifications of methods from the theories of several complex variables and complex analytic geometry and would serve as excellent training for mathematicians wanting to work in complex analytic geometry.simply{connected Riemann surface is isomorphic to Cb, C or H.

Let us consider each of these in turn. The Riemann sphere. The simplest compact Riemann surface is Cb= C[1with charts U 1 = C and U 2 = Cbf 0gwith f 1(z) = zand f 2(z) = 1=z. Alternatively, Cb˘=P1 is the space of lines in C2: P1 = (C2 0)=C: The isomorphism is given by [Z 0: Z 1] 7!z.Every compact Riemann surface admits a holomorphic embedding into P3.

(See [1] page ) A closed holomorphic submanifold of PN is a smooth algebraic variety (Chow’s Theorem, see [2] page ); hence every Riemann surface is isomorphic to a smooth algebraic curve. Let C PN be an algebraic curve and S Cbe the set of singular points.